\documentclass[12pt,a4paper,pdftex]{exam}

\usepackage[english]{babel}
\usepackage{pslatex}
\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}

%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
\pagestyle{headandfoot}
\ifprintanswers
\newcommand{\stitle}{: Solutions}
\else
\newcommand{\stitle}{}
\fi
\header{{\bfseries\large Exercise 9\stitle}}{{\bfseries\large Bootstrap}}{{\bfseries\large December 9th, 2019}}
\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
jan.benda@uni-tuebingen.de}
\runningfooter{}{\thepage}{}

\setlength{\baselineskip}{15pt}
\setlength{\parindent}{0.0cm}
\setlength{\parskip}{0.3cm}
\renewcommand{\baselinestretch}{1.15}

%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{listings}
\lstset{
  language=Matlab,
  basicstyle=\ttfamily\footnotesize,
  numbers=left,
  numberstyle=\tiny,
  title=\lstname,
  showstringspaces=false,
  commentstyle=\itshape\color{darkgray},
  breaklines=true,
  breakautoindent=true,
  columns=flexible,
  frame=single,
  xleftmargin=1em,
  xrightmargin=1em,
  aboveskip=10pt
}

%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm} 
\usepackage{dsfont}
\newcommand{\naZ}{\mathds{N}}
\newcommand{\gaZ}{\mathds{Z}}
\newcommand{\raZ}{\mathds{Q}}
\newcommand{\reZ}{\mathds{R}}
\newcommand{\reZp}{\mathds{R^+}}
\newcommand{\reZpN}{\mathds{R^+_0}}
\newcommand{\koZ}{\mathds{C}}

%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\continue}{\ifprintanswers%
\else
\vfill\hspace*{\fill}$\rightarrow$\newpage%
\fi}
\newcommand{\continuepage}{\ifprintanswers%
\newpage
\else
\vfill\hspace*{\fill}$\rightarrow$\newpage%
\fi}
\newcommand{\newsolutionpage}{\ifprintanswers%
\newpage%
\else
\fi}

%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\qt}[1]{\textbf{#1}\\}
\newcommand{\pref}[1]{(\ref{#1})}
\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
\newcommand{\code}[1]{\texttt{#1}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\input{instructions}

\begin{questions}

\question \qt{Bootstrap the standard error of the mean}
We want to compute the standard error of the mean of a data set by
means of the bootstrap method and compare the result with the formula
``standard deviation divided by the square-root of $n$''.
\begin{parts}
  \part Download the file \code{thymusglandweights.dat} from Ilias.
  This is a data set of the weights of the thymus glands of 14-day old chicken embryos
  measured in milligram.
  \part Load the data into Matlab (\code{load} function).
  \part Compute histogram, mean, and standard error of the mean of the first 80 data points.
  \part Compute the standard error of the mean of the first 80 data
  points by bootstrapping the data 500 times. Write a function that
  bootstraps the standard error of the mean of a given data set. The
  function should also return a vector with the bootstrapped means.
  \part Compute the 95\,\% confidence interval for the mean from the
  bootstrap distribution (\code{quantile()} function) --- the
  interval that contains the true mean with 95\,\% probability.
  \part Use the whole data set and the bootstrap method for computing
  the dependence of the standard error of the mean from the sample
  size $n$.
  \part Compare your result with the formula for the standard error
  $\sigma/\sqrt{n}$.
\end{parts}
\begin{solution}
  \lstinputlisting{bootstrapmean.m}
  \lstinputlisting{bootstraptymus.m}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
\end{solution}


\question \qt{Student t-distribution}
The distribution of Student's t, $t=\bar x/(\sigma_x/\sqrt{n})$, the
estimated mean $\bar x$ of a data set of size $n$ divided by the
estimated standard error of the mean $\sigma_x/\sqrt{n}$, where
$\sigma_x$ is the estimated standard deviation, is not a normal
distribution but a Student-t distribution.  We want to compute the
Student-t distribution and compare it with the normal distribution.
\begin{parts}
\part Generate 100000 normally distributed random numbers.
\part Draw from these data 1000 samples of size $n=3$, 5, 10, and
50. For each sample size $n$ ...
\part ... compute the mean $\bar x$ of the samples and plot the
probability density of these means.
\part ... compare the resulting probability densities with corresponding
normal distributions.
\part ... compute Student's $t=\bar x/(\sigma_x/\sqrt{n})$ and compare its
distribution with the normal distribution with standard deviation of
one. Is $t$ normally distributed? Under which conditions is $t$
normally distributed?
\end{parts}
\newsolutionpage
\begin{solution}
  \lstinputlisting{tdistribution.m}
  \includegraphics[width=1\textwidth]{tdistribution-n03}\\
  \includegraphics[width=1\textwidth]{tdistribution-n05}\\
  \includegraphics[width=1\textwidth]{tdistribution-n10}\\
  \includegraphics[width=1\textwidth]{tdistribution-n50}
\end{solution}


\continue
\question \qt{Permutation test} \label{permutationtest}
We want to compute the significance of a correlation by means of a permutation test.
\begin{parts}
  \part \label{permutationtestdata} Generate 1000 correlated pairs
  $x$, $y$ of random numbers according to:
\begin{verbatim}
n = 1000
a = 0.2;
x = randn(n, 1);
y = randn(n, 1) + a*x;
\end{verbatim}
  \part Generate a scatter plot of the two variables.
  \part Why is $y$ correlated with $x$?
  \part Compute the correlation coefficient between $x$ and $y$.
  \part What do you need to do in order to destroy the correlations between the $x$-$y$ pairs?
  \part Do exactly this 1000 times and compute each time the correlation coefficient.
  \part Compute and plot the probability density of these correlation
  coefficients.
  \part Is the correlation of the original data set significant?
  \part What does ``significance of the correlation'' mean?
%  \part Vary the sample size \code{n} and compute in the same way the
%  significance of the correlation.
\end{parts}
\begin{solution}
  \lstinputlisting{correlationsignificance.m}
  \includegraphics[width=1\textwidth]{correlationsignificance}
\end{solution}

\question \qt{Bootstrap the correlation coefficient} 
The permutation test generates the distribution of the null hypothesis
of uncorrelated data and we check whether the correlation coefficient
of the data differs significantly from this
distribution. Alternatively we can bootstrap the data while keeping
the pairs and determine the confidence interval of the correlation
coefficient of the data. If this differs significantly from a
correlation coefficient of zero we can conclude that the correlation
coefficient of the data indeed quantifies correlated data.

We take the same data set that we have generated in exercise
\ref{permutationtest} (\ref{permutationtestdata}).
\begin{parts}
  \part Bootstrap 1000 times the correlation coefficient from the data.
  \part Compute and plot the probability density of these correlation
  coefficients.
  \part Is the correlation of the original data set significant?
\end{parts}
\begin{solution}
  \lstinputlisting{correlationbootstrap.m}
  \includegraphics[width=1\textwidth]{correlationbootstrap}
\end{solution}

\end{questions}

\end{document}